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Consider the equation:
|x-5|2 + 5 |x - 5| - 24 = 0
The sum of all the real roots of the above equationis:
- a)10
- b)3
- c)8
- d)2
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Consider the equation:|x-5|2 + 5 |x - 5| - 24 = 0The sum of all the re...
Let's consider x-5 as 'p'
Case 1: p ≥ 0
|x - 5| |2 + 5|x - 5| - 24 = 0
p2 +5p - 24 = 0
p2 + 8p - 3p - 24 = 0
p(p + 8) -3 (p + 8) = 0
(p + 8) (p - 3) = 0
p = -8 and p = 3
x - 5 = 3,x = 8 This is a real root since x is greater than 5.
x - 5 = -8, x = -3. This root can be negated because x is not greater than 5.
Case 2: p < 0
p2 - 5p - 24 = 0
p2 - 8p + 3p - 24 = 0
p=8, -3
x - 5 = 8, x = 13. This root can be negated because x is not less than 5
x - 5 = -3, x = 2. This is a real root because x is less than 5.
The sum of the real roots = 8 + 2 = 10
Most Upvoted Answer
Consider the equation:|x-5|2 + 5 |x - 5| - 24 = 0The sum of all the re...
Given Equation:
The equation given is |x-5|2 + 5 |x - 5| - 24 = 0.
Finding the Real Roots:
To find the real roots of the equation, we can substitute y = |x - 5| and rewrite the equation as y^2 + 5y - 24 = 0.
Solving the Quadratic Equation:
Now, we can solve the quadratic equation y^2 + 5y - 24 = 0 by factoring or using the quadratic formula.
The factors of -24 that add up to 5 are 8 and -3. Therefore, the equation can be factored as (y + 8)(y - 3) = 0.
So, the solutions for y are y = -8 and y = 3.
Finding the Roots of the Original Equation:
Since y = |x - 5|, we substitute y back in and solve for x to find the real roots.
For y = -8, x - 5 = -8 which gives x = -3.
For y = 3, x - 5 = 3 which gives x = 8.
Sum of Real Roots:
The sum of the real roots of the equation is -3 + 8 = 5.
Therefore, the correct answer is option 'A' which is 5.
The equation given is |x-5|2 + 5 |x - 5| - 24 = 0.
Finding the Real Roots:
To find the real roots of the equation, we can substitute y = |x - 5| and rewrite the equation as y^2 + 5y - 24 = 0.
Solving the Quadratic Equation:
Now, we can solve the quadratic equation y^2 + 5y - 24 = 0 by factoring or using the quadratic formula.
The factors of -24 that add up to 5 are 8 and -3. Therefore, the equation can be factored as (y + 8)(y - 3) = 0.
So, the solutions for y are y = -8 and y = 3.
Finding the Roots of the Original Equation:
Since y = |x - 5|, we substitute y back in and solve for x to find the real roots.
For y = -8, x - 5 = -8 which gives x = -3.
For y = 3, x - 5 = 3 which gives x = 8.
Sum of Real Roots:
The sum of the real roots of the equation is -3 + 8 = 5.
Therefore, the correct answer is option 'A' which is 5.
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